An Example in the Singer Category of Algebras with Coproducts at Odd Primes

Abstract

In 2005, William M. Singer introduced the notion of k-algebra with\ud coproducts for any commutative ring k, and showed that the algebra of operations\ud on the cohomology ring of any cocommutative F2-Hopf algebra can be endowed\ud with such structure. In this paper we show that the same is true when the ground\ud field of the cocommutative Hopf algebra is Fp, p is any odd prime, and the algebra\ud of operations B(p) is equipped with an exotic coproduct. We also give an explicit\ud description of the coalgebra with products dual to B(p)

A note on the algebra of operations for Hopf cohomology at odd primes

“…As already noted in [7], Item (iii) of Definition 2.1 makes sense: the category of graded coalgebras has tensor products and sums, and the category of graded algebras has tensor products and categorical products. Explicitly, given two graded algebras A and B, on A b B we assume defined the product…”

“…Along the spirit of [25], in [7] the authors equipped Bppq with a suitable collection of F p -linear mappings that made it the underlying set of an object in the Singer category of F p -algebras with coproducts. Yet, in [7], the chosen coproduct acting on the Bockstein operator β has little to do with its nature of cohomology operation.…”

Length-preserving monomorphisms for some algebras of operations

“…Along the proof of Proposition 9 in [7], it is proved that the algebra B( p) of Steenrod operations on Ext H (F p , F p ), the cohomology of a graded cocommutative F p -Hopf algebra H , is the direct sum (as graded vector space) of two isomorphic summands, the former being also a subalgebra and isomorphic to C( p). Section 3 ends by computing Aut(L 0 ( p)).…”

“…In [20] Peter May introduced an F p -algebra Q( p) known by different names such as the algebra of all generalized Steenrod operations [19] and the universal Steenrod algebra (see [2][3][4][5][6][7][8][9][10][11][12][13][15][16][17][18]). It is the algebra of cohomology operations in the category C( p, ∞) of H ∞ -ring spectra.…”

Searching for Fractal Structures in the Universal Steenrod Algebra at Odd Primes

“…The algebra Q(p) is related to many Steenrod-like operations. For instance to those acting on the cohomology of a graded cocommutative Hopf algebra ( [6], [14]), or the Dyer-Lashof operations on the homology of infinite loop spaces ( [1] and [17]). Details of such connections, at least for p = 2, can be found in [5].…”